Learning Sensorimotor Agency
in Cellular Automata

Finding robust self-organizing "agents" with gradient descent and curriculum learning: individuality, self-maintenance and sensori-motricity within a cellular automaton environment

Interactive Demo


Zoomed in

Multi creature



Video S1: Sensorimotor agents
Different agents (yellow) emerging from rules obtained by the IMGEP. The agents display sensorimotor capabilities: they are robust and react to perturbations by the obstacles (blue). The righmost video shows the system with a different colormap (fixed obstacle channel in black) to highlight the differences in activity in the agent as a response to perturbation.

Video S2: Random search
Each 100 squares are random parameters trials (each 1 channel and 10 rules so \(\sim \) 130 parameters for all the rules of a square). We observe that a lot of random search trials lead to death or explosion of the mass. Very little lead to stable spatially localized pattern and even less to moving ones.

Video S3: Orbium, moving agent from the original lenia papers, fragile to external perturbations
Left: Orbium: the equivalent of the glider in Lenia (from the original lenia paper), an example of moving agent. Middle and right videos: collision between several orbium leading to death/explosion. This shows the fragility of the orbium to external perturbations.

Video S4: Orbium perturbed by obstacles.
Orbium, equivalent of the glider in Lenia (from the original lenia paper), dies from perturbations by obstacles.

Random search

IMGEP search

Handmade search

Video S5: Agents obtained by each method
100 Patterns passing our agency tests obtained by each method (from left to right): random search,IMGEP, handmade search (from Lenia original papers). A lot of IMGEP obtained agents are moving agents with high speed while a lot of agents obtained by random search are static.


Random search

IMGEP search

Handmade search

Video S6: Moving obstacle test on agents obtained by each method
100 Agents obtained by random search,IMGEP, handmade search (from Lenia original papers). We observe that the proportion of agents with robustness to moving obstacles is much higher in the agents obtained by IMGEP than the ones obtained by random search and handmade search.


Seed 9 IMGEP step 142

Video S7.1
Obstacle number : 30

Video S7.2
Obstacle radius 7

Video S7.3
Obstacle speed 3

Video S7.4
Update mask rate 0.2

Video S7.5
Update noise rate 0.8

Video S7.6
Update noise std 0.6

Video S7.7
Init noise rate 1

Video S7.8
Init noise std 4.5

Video S7.9
Rescaling 2.15

Seed 5 IMGEP step 120

Video S7.11
Obstacle number : 30

Video S7.12
Obstacle Radius 7.

Video S7.13
Obstacle speed 3

Video S7.14
Update mask rate 0.2

Video S7.15
Update noise rate 0.8

Video S7.16
Update noise std 0.6

Video S7.17
Init noise rate 1.

Video S7.18
Init noise std 4.5

Video S7.19
Rescaling 2.15

Video S7 : Illustration of the quantitative tests performed
Videos of quantitative tests for 2 moving agents obtained by IMGEP. We display only a subset of the value tested for every quantitative test.


Video S8: Out of distribution obstacles: Different shapes
Test of a moving agent obtained by IMGEP on obstacles that were not seen during training.

Video S9: Out of distribution obstacles: maze.
Test of a moving agent to maze like obstacles.

Video S10: Out of distribution obstacles: Bullet like obstacles
Test of a moving agent to bullet like environment: fast small moving obstacles.

Video S11: Individuality preservation
Example of moving agents obtained by IMGEP colliding while keeping their individuality, they don't merge or collapse from the collision.

Video S12: Reproduction
For some moving agents, under specific conditions, the collision of 2 agents can lead to the self-organization of a 3rd agent. (each with its own individuality)

Video S13: Attraction
Example of moving agents attracting each other while still maintaining their own individuality.

Video S14: Asynchronous update
Testing a moving agent with asynchronous updates. Each cell is updated with a certain probability at each step leading to cells being asynchronously updated.

Video S15: Scaling the agents down
The moving agents size is reduced. The scaled down agents still seem to behave similarly (same shape and have sensorimotor capabilities) to the normal size one while being composed of less cells.

Video S16: Morphological computation
We pause the simulation and remove some cells of a moving agent. As a response to this alteration of the structure, the moving agent changes direction, regrow itself and moves away. This video isolates the fact that the macro agent senses perturbations of its structure and respond to it by a morphological growth.

Video S17: External control.
We introduce an attractive element in another channel (in Cyan). We learned the rule that control the way this external element channel influences the learnable channel(Yellow) and display the resulting behavior here. The moving agent is effectively attracted to this introduce component. By controlling the external element we can control live the direction of the moving agent.

Video S18: Robustness to initialization
Testing the robustness of the learned rule to emerge an agent from different initial patterns. We replace the learned initial pattern by : left a disk with a gradient; middle a large disk (much larger than an agent); right right top a disk with gradient of another size, right bottom a disk without gradient. Some initialization lead to the robust emergence of one or several agents while some lead to the collapse of the pattern.


Video S19: Examples of agents considered non moving by our moving test

Video S20: Unstability
Example of partial robustness of a moving agent. The agent is able to deal with some perturbation by obstacle but some other might break the equilibrium (leading to explosion)

Video S21: Visualization of each rule
For the visualization of the role of each rule, we display the growth (in the magma colormap) of each of the 10 rules in the exterior videos. The inner top video shows the normal simulation. The inner bottom video displays the weighted sum of the growth of all the rules in grey level (1 being white and 0 being black), showing what is added to the current state of the learnable channel.